The inner surfaces of the ring electrode and the end cap electrodes of an ion trap mass spectrometer are shaped hyperboloids, having a hyperbolic lateral surface in their central cross section. When an appropriate voltage is applied to these electrodes, an electric field is generated in the space surrounded by these electrodes which provides the analyzing space of the mass spectrometer. The electric field, .phi.(r,z), is ideally represented by the following quadrupole electric field as: EQU .phi.(r,z) r.sup.2 -2z.sup.2 (1),
where r and z are the coordinates of the cylindrical coordinate system with r denoting the distance from the central axis of the ion trap toward the ring electrode, and z denoting the distance from the center of the ion trap toward an end cap electrode.
When an RF (radio frequency) voltage V of frequency .OMEGA. is applied to the ring electrode with a DC (direct current) voltage U superposed, ions are trapped in the analyzing space of the quadrupole electric field generated therein. The ion trapping condition is determined by various parameters including the RF voltage V, the frequency .OMEGA., the DC voltage U, and the dimensions of the apparatus (the radius r.sub.0 of the ring electrode and the half distance z.sub.c between the end cap electrodes).
The ion trapping condition is represented, for example, by the q.sub.z -a.sub.z plane as shown In stability diagram of FIG. 14. The equation of motion for an ion having mass m and electric charge e is given by the generalized Mathieu equation as: EQU d.sup.2 u/d.xi..sup.2 +(a.sub.n -2.multidot.q.sub.n .multidot.cos(2.multidot..xi.)).multidot.u=0 (2),
where EQU u=x, y, z (3), EQU .xi.=.OMEGA..multidot.t/2 (4), EQU a.sub.z =-2.multidot.a.sub.x =-2.multidot.a.sub.y =-8.multidot.e.multidot.U/(m.tau..sub.0.sup.2 .multidot..OMEGA..sup.2)(5),
and EQU q.sub.z =-2.multidot.q.sub.x =-2.multidot.q.sub.y =4.multidot.e.multidot.V/(m.tau..sub.0.sup.2 .multidot..OMEGA..sup.2)(6).
The parameters a.sub.z and q.sub.z are determined by the mass to charge ratio m/e of the ion. When a set of parameters (a.sub.z, q.sub.z) lies within the stability region as shown in FIG. 14, an ion of corresponding m/e oscillates at a certain frequency, which is called the secular frequency, and Is trapped in the analyzing space. The parameter .beta. in FIG. 14 is a value depending on the parameter q.
In an ion trap mass spectrometer, a mass spectrum is obtained through a method using the mass-selective instability scan mode in which ions are ejected through one or a plurality of holes formed at the center of an end cap electrode and are detected while the RF voltage V is continuously increased. When RF voltage is solely applied to the electrodes, a.sub.z is zero (a.sub.x =0) and q.sub.z has a certain value depending on the m/e ratio of the ion. As the RF voltage is increases, q.sub.z increases correspondingly. When a set of parameters (a.sub.z, q.sub.z) approaches the boundary of the stability region (a.sub.z =0, q.sub.z =0.908), oscillation of ions along the z direction becomes unstable, and ions are ejected through the hole or holes of the end cap electrode. This means that the RF voltage where ions are ejected is proportional to the m/e ratio, and a mass spectrum is obtained scanning the RF voltage V as a parameter representative of the m/e ratio.
Another method of obtaining a mass spectrum in an ion trap mass spectrometer is the resonance ejection mode in which, similarly to the previous method, a mass spectrum is obtained while the RF voltage is continuously increased. An auxiliary AC (alternating current) voltage is applied between the end cap electrodes. When the frequency of the auxiliary AC voltage coincides with the secular frequency of ions, the AC voltage excites a resonance oscillation of the ions and ejects them from the analyzing space. Thus a mass spectrum is obtained through ejection of ions at the frequency of the auxiliary AC voltage because the secular frequencies of ions are determined by the parameters a.sub.z and q.sub.z and successively match the frequency with increasing RF voltage.
Since electrodes of an actual ion trap mass spectrometer must have finite dimensions, the theoretically infinite hyperbolic surface should be truncated at a finite extent. This causes a deviation of the actual electric field from a pure quadrupole electric field as used in the theory and deteriorates the performance of the mass spectrometer. The direction of the deviation in the peripheral region of the analyzing space tends to a lower electric field than a pure quadrupole electric field. When the electric field in the analyzing space is represented by multipole expansion, the signs of the quadrupole component and the sum of the other multipole components (hexapole and octopole, for example) are opposite.
This deviation reduces the force acting on the ions when the z-directional oscillation becomes unstable and the amplitude of the oscillation is increasing, at around q.sub.z =0.908 in the mass-selective instability scan mode, compared to the case of using a pure quadrupole electric field. The reduction of the force is regarded as a reduction of the effective RF voltage, and of q.sub.z, and the ion is pulled back into the stability region. This requires further increase of the RF voltage to eject the ions causing deterioration of performance, such as mass resolution. A similar problem is observed in the resonance ejection mode.
The deviation from a pure quadrupole field introduced by truncation of the electrodes can he alleviated by extending the position of the truncation but the deviation of the electric field still has an opposite sign to a pure quadrupole electric field. The aforementioned problem, the deterioration of the performance, can not be solved by this means.
Two methods are conventionally used to solve the problem. One is a method using a stretched geometry mode of the electrodes in which the end cap electrodes are separated further apart than the theoretically determined positions, as shown in FIG. 15. The other method is shown in FIG. 16 in which the surfaces of the ring electrode and the end cap electrodes are deviated from the theoretically required position so that the asymptotes are slightly skewed. The solid lines show theoretical positions of the asymptotes and dotted lines show their modifications in FIGS. 15 and 16. The two methods correct the deviations of the electric field by superposing electric fields of the same polarity as the quadrupole electric field throughout the analyzing space.